Group algebras¶
This functionality has been moved to sage.categories.algebra_functor
.
- sage.algebras.group_algebra.GroupAlgebra(G, R=Integer Ring)¶
Return the group algebra of \(G\) over \(R\).
INPUT:
\(G\) – a group
\(R\) – (default: \(\ZZ\)) a ring
EXAMPLES:
The group algebra \(A=RG\) is the space of formal linear combinations of elements of \(G\) with coefficients in \(R\):
sage: G = DihedralGroup(3) sage: R = QQ sage: A = GroupAlgebra(G, R); A Algebra of Dihedral group of order 6 as a permutation group over Rational Field sage: a = A.an_element(); a () + (1,2) + 3*(1,2,3) + 2*(1,3,2)
This space is endowed with an algebra structure, obtained by extending by bilinearity the multiplication of \(G\) to a multiplication on \(RG\):
sage: A in Algebras True sage: a * a 14*() + 5*(2,3) + 2*(1,2) + 10*(1,2,3) + 13*(1,3,2) + 5*(1,3)
GroupAlgebra()
is just a short hand for a more general construction that covers, e.g., monoid algebras, additive group algebras and so on:sage: G.algebra(QQ) Algebra of Dihedral group of order 6 as a permutation group over Rational Field sage: GroupAlgebra(G,QQ) is G.algebra(QQ) True sage: M = Monoids().example(); M An example of a monoid: the free monoid generated by ('a', 'b', 'c', 'd') sage: M.algebra(QQ) Algebra of An example of a monoid: the free monoid generated by ('a', 'b', 'c', 'd') over Rational Field
See the documentation of
sage.categories.algebra_functor
for details.
- class sage.algebras.group_algebra.GroupAlgebra_class(R, basis_keys=None, element_class=None, category=None, prefix=None, names=None, **kwds)¶